I have an integral of the form $$ I=\int_0^{\infty}{dx}\ln \bigg(1+\exp(-\frac{f(x)}{a})\bigg) $$ where $a$ is a positive constant and $f(x)$ is a regular and positive function such that $I$ is finite (for example: $f(x)=x$); moreover $f(x)=x+O(x^3)$ for $x \rightarrow 0$. I have to expand this integral $I=c_0+c_1a+..+c_na^n+O(a^{n+1})$ for $a \rightarrow 0^+$, with the value of the integral beeing dominated by the behaviour of $f(x)$ near $0$. Can anyone give me some hints or references to compute the first coefficients of this expansion?

  • $\begingroup$ Does the integral converge at infinity? Does not look so. $\endgroup$ – Fedor Petrov Feb 21 at 17:24
  • $\begingroup$ I've edited the question, $f(x)$ is such that $I$ is finite $\endgroup$ – John Feb 21 at 17:30
  • $\begingroup$ how can it be finite? $\endgroup$ – Fedor Petrov Feb 21 at 18:26
  • $\begingroup$ It is an hypothesis. $\endgroup$ – John Feb 21 at 18:29
  • $\begingroup$ I've edited my question: f(x) is positive and $I$ finite. An example: $f(x)=x$. The point is: how can i do the expansion? I had in mind this example: f(x)=x*g(x), where g(x) is posivite and bounded $\endgroup$ – John Feb 21 at 18:36

Let $f(x)$ have the expansion $f(x)=x+c_3 x^3+c_4 x^4 +\cdots$, then define $y=x/a$ and you have $$I=\int_0^{\infty}{dx}\ln \bigg(1+\exp\left(-\frac{f(x)}{a}\right)\bigg)$$ $$=\int_0^\infty \left[a\ln \left(1+e^{-y}\right)-\frac{a^3 c_3 y^3}{e^y+1}-\frac{a^4 c_4 y^4}{e^y+1}+{\cal O}(a^5)\right]\,dy$$ $$=\frac{1}{12} \pi ^2 a-\frac{7}{120} \pi ^4 c_3 a^3-\frac{45}{2} \zeta (5)c_4 a^4+{\cal O}(a^5).$$ The term of order $a^p$ has coefficient $-c_p\left(1-2^{-p}\right)p! \zeta (p+1)$, when $c_p$ is the coefficient of order $x^p$ in the expansion of $f(x)$.

  • $\begingroup$ Reminiscent of "Watson's Lemma"? $\endgroup$ – paul garrett Feb 21 at 19:18
  • $\begingroup$ Thank you for the answer. Could you give some details on how you obtain the expansion of the integrand? $\endgroup$ – John Feb 21 at 20:27
  • $\begingroup$ @paul garrett-- Could you give some more details on this Lemma? $\endgroup$ – John Feb 21 at 21:14
  • $\begingroup$ @John, google will find Watson's lemma, for example. I don't remember where I first saw it, but it appears early on in any book on "asymptotic expansions" (another keyword). I think Erdelyi's book does this, for example. I wrote up something with bibliographic pointers: linked to from my "intro to modular forms" page (but not depending on that stuff) at math.umn.edu/~garrett/m/mfms, under label "asymptotics of integrals somewhere on the page", precise link math.umn.edu/~garrett/m/mfms/notes_2013-14/… $\endgroup$ – paul garrett Feb 21 at 21:44
  • $\begingroup$ en.wikipedia.org/wiki/Watson%27s_lemma $\endgroup$ – Carlo Beenakker Feb 21 at 21:48

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